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A positive eigenvalue result for semilinear differential equations in Banach spaces with functional initial conditions

Gennaro Infante, Paola Rubbioni

TL;DR

This work establishes a general, semigroup-based framework for the existence of positive eigenvalues in semilinear differential equations on Banach spaces with functional (possibly nonlocal) initial data. By formulating mild solutions via $y(t)=\lambda U(t)B(y)+\lambda\int_0^t U(t-s)f(s,y(s))\, ds$ and employing a cone-based Krasnoselskii-type fixed-point argument, the authors prove that, under conditions $(H_1)-(H_4)$, for each radius $\rho>0$ there exists a positive eigenpair $(\lambda_\rho,y_\rho)$ with $\|y_\rho\|_C=\rho$. This implies, when $\rho$ varies, uncountably many eigenpairs, providing a robust tool for nonlocal initial-value problems. The theory is exemplified with a reaction–diffusion equation featuring a nonlocal initial condition, illustrating applicability to PDE models with nonlocal data arising in heat-flow and sensor-based settings.

Abstract

We study the existence of positive eigenvalues with associated nonnegative mild eigenfunctions for a class of abstract initial value problems in Banach spaces with functional, possibly nonlocal, initial conditions. The framework includes periodic, multipoint, and integral average conditions. Our approach relies on nonlinear analysis, topological methods, and the theory of strongly continuous semigroups, yielding results applicable to a wide range of models. As an illustration, we apply the abstract theory to a reaction-diffusion equation with a nonlocal initial condition arising from a heat flow problem.

A positive eigenvalue result for semilinear differential equations in Banach spaces with functional initial conditions

TL;DR

This work establishes a general, semigroup-based framework for the existence of positive eigenvalues in semilinear differential equations on Banach spaces with functional (possibly nonlocal) initial data. By formulating mild solutions via and employing a cone-based Krasnoselskii-type fixed-point argument, the authors prove that, under conditions , for each radius there exists a positive eigenpair with . This implies, when varies, uncountably many eigenpairs, providing a robust tool for nonlocal initial-value problems. The theory is exemplified with a reaction–diffusion equation featuring a nonlocal initial condition, illustrating applicability to PDE models with nonlocal data arising in heat-flow and sensor-based settings.

Abstract

We study the existence of positive eigenvalues with associated nonnegative mild eigenfunctions for a class of abstract initial value problems in Banach spaces with functional, possibly nonlocal, initial conditions. The framework includes periodic, multipoint, and integral average conditions. Our approach relies on nonlinear analysis, topological methods, and the theory of strongly continuous semigroups, yielding results applicable to a wide range of models. As an illustration, we apply the abstract theory to a reaction-diffusion equation with a nonlocal initial condition arising from a heat flow problem.
Paper Structure (4 sections, 6 theorems, 92 equations)

This paper contains 4 sections, 6 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem setting and main result
  4. An application to reaction-diffusion equations

Key Result

Theorem 2.1

Krasno Let $(X, \| \cdot \|)$ be a real Banach space, $U \subset X$ an open bounded set with $0 \in U$, ${\cal K} \subset X$ a cone, and let $T \colon {\cal K} \cap \bar{U} \to {\cal K}$ be a compact operator (i.e. it is continuous and maps bounded sets into relatively compact sets). Suppose that Then there exist $\lambda_0 \in (0, +\infty)$ and $x_0 \in {\cal K} \cap \partial U$ such that $x_0 =

Theorems & Definitions (9)

  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 3.1
  • Theorem 3.1
  • Corollary 3.1
  • Theorem 4.1
  • Example 4.1